Can someone please check my answers? Thanks! :)
1.) Which of the following inequalities is true for all real values of x?
a.) x^2≤x^3
b.) 2x^3≤4x^2
c.) 4x^2≤94x)^2
d.) 4(x-3)^2≥4x^2-3
My answer: B.
2.) Which of the following inequalities is true for all real values of x?
a.) (5x)^2≥5x^2
b.) x^6≤x^7
c.) 5x^2≥2x^3
d.) 6(x-5)^2≥6x^2-5
My answer: D.
Answer this Q
#1, Clearly not true for x=10
Try C if it's supposed to read
4x^2 <= (4x)^2
because (4x)^2 = 16x^2
#2 A, just as in #1.
(5x)^2 = 25x^2 >= 5x^2
take a look at wolframalpha.com, where you can plot several functions on the same axes. For example,
http://www.wolframalpha.com/input/?i=+plot+y%3D5x^2%2C+y%3D2x^3
To check the answers for these inequalities, we can go through each option and substitute some real values of x to see which inequalities hold true.
For the first question:
a.) x^2 ≤ x^3
To check this inequality, substitute x = 1.
1^2 ≤ 1^3
1 ≤ 1
This inequality holds true for x = 1.
b.) 2x^3 ≤ 4x^2
To check this inequality, substitute x = 1.
2(1)^3 ≤ 4(1)^2
2 ≤ 4
This inequality holds true for x = 1.
c.) 4x^2 ≤ 94x)^2
This option seems to have a typo. It should be 4x^2 ≤ 9(4x)^2 for it to make sense. Assuming this was the intended inequality, we can check it by substituting x = 1.
4(1)^2 ≤ 9(4(1))^2
4 ≤ 9(4)^2
4 ≤ 9(16)
4 ≤ 144
This inequality holds true for x = 1.
d.) 4(x-3)^2 ≥ 4x^2-3
To check this inequality, substitute x = 1.
4(1-3)^2 ≥ 4(1)^2-3
4(-2)^2 ≥ 4-3
4(4) ≥ 1
16 ≥ 1
This inequality holds true for x = 1.
Based on our analysis, all four options hold true for x = 1. Therefore, your answer of B is incorrect. The correct answer is all of the above options: a, b, c, and d.
Moving on to the second question:
a.) (5x)^2 ≥ 5x^2
To check this inequality, substitute x = 1.
(5(1))^2 ≥ 5(1)^2
5^2 ≥ 5
25 ≥ 5
This inequality holds true for x = 1.
b.) x^6 ≤ x^7
To check this inequality, substitute x = 1.
1^6 ≤ 1^7
1 ≤ 1
This inequality holds true for x = 1.
c.) 5x^2 ≥ 2x^3
To check this inequality, substitute x = 1.
5(1)^2 ≥ 2(1)^3
5 ≥ 2
This inequality holds true for x = 1.
d.) 6(x-5)^2 ≥ 6x^2-5
To check this inequality, substitute x = 1.
6(1-5)^2 ≥ 6(1)^2-5
6(-4)^2 ≥ 6-5
6(16) ≥ 1
96 ≥ 1
This inequality holds true for x = 1.
Based on our analysis, all four options hold true for x = 1. Therefore, your answer of D is incorrect. The correct answer is all of the above options: a, b, c, and d.